-Tamari lattices via subword complexes Cesar Ceballos ( joint with - - PowerPoint PPT Presentation

ν-Tamari lattices via subword complexes

Cesar Ceballos (joint with Arnau Padrol and Camilo Sarmiento)

The 78th S´ eminaire Lotharingien de Combinatoire Ottrott, March 28, 2016

In this talk

Theorem

The ν-Tamari lattice is the dual of a well chosen subword complex.

1 1 1 1 1 1 1 2 2 1 2 1 1 1 1 1 1 1 1 2 2 2 2 2 2 2 2 2 2 2 2 2

(0,0,1,1,1,2) (1,0,1,1,1,2) (2,0,1,1,1,2) (2,0,2,1,1,2) (2,0,2,2,1,2) (0,0,2,2,1,2) (0,0,2,1,1,2)

In this talk

Theorem

The ν-Tamari lattice is the dual of a well chosen subword complex.

1 1 1 1 1 1 1 2 2 1 2 1 1 1 1 1 1 1 1 2 2 2 2 2 2 2 2 2 2 2 2 2

(0,0,1,1,1,2) (1,0,1,1,1,2) (2,0,1,1,1,2) (2,0,2,1,1,2) (2,0,2,2,1,2) (0,0,2,2,1,2) (0,0,2,1,1,2)

The picture actually contains three theorems and one corollary. Please remember the picture!

Tamari lattices

The Tamari-lattice: partial order on Catalan objects.

• Tamari. Mono¨

ıdes pr´ eordonn´ es et chaˆ ınes de Malcev. Doctoral Thesis, Paris 1951. Associahedra, Tamari Lattices and Related Structures. Birkh¨ auser/Springer, 2012.

Tamari lattices

The Tamari-lattice is a partial order on Catalan objects. Covering relation:

A B C A B C

Rotation on binary trees

Tamari lattices

The Tamari-lattice is a partial order on Catalan objects. Covering relation:

Interchanging operation on Dyck paths

m-Tamari lattices

Motivated by trivariate diagonal harmonics, F. Bergeron Introduced the m-Tamari lattice on Fuss-Catalan paths.

• F. Bergeron–Pr´

eville-Ratelle. Higher trivariate diagonal harmonics via generalized Tamari posets. J. Comb 3(3), 2012.

m-Tamari lattices: nice enumerative properties

◮ Number of elements: Fuss Catalan number

1 mn+1

(m+1)n

n

• ◮ Number of intervals:

m+1 n(mn+1)

(m+1)2n+m

n−1

• Chapoton. Sur le nombre d’intervalles dans les treillis de Tamari. S´
• em. Lothar.

Combin., 55, 2005/07. (m=1)

• F. Bergeron–Pr´

eville-Ratelle. Higher trivariate diagonal harmonics via generalized Tamari posets. J. Comb 3(3), 2012. (conjectured) Bousquet-M´ elou–Fusy–Pr´ eville-Ratelle. The number of intervals in the m-Tamari lattices. Electron. J. Combin., 18(2), 2011. (proof) ◮ Number of “decorated” intervals: (m + 1)n(mn + 1)n−2 Bousquet-M´ elou–Chapuy–Pr´ eville-Ratelle. The representation of the symmetric group on m-Tamari intervals. Adv. Math., 2013.

Conjecture (F. Bergeron (Haiman for m = 1))

The number of intervals is conjecturally interpreted as the dimension of the alternating component of a space in trivariate diagonal harmonics. Decorated intervals correspond to the entire space.

m-Tamari lattices: nice geometry

The 2-Tamari lattice for n = 4

C.–Padrol–Sarmiento, 2016: The Hasse diagram of m-Tamari lattices are the edge graphs of (tropical) polytopal subdivisions

• f associahedra.

ν-Tamari lattices

Pr´ eville-Ratelle–Viennot: Introduced the ν-Tamari lattice on lattice paths weakly above ν. Covering relation:

Theorem (Pr´ eville-Ratelle–Viennot)

This partial order defines a lattice structure on ν-Dyck paths.

Pr´ eville-Ratelle–Viennot. An extension of Tamari lattices. To appear in Trans. AMS.

ν-Tamari lattices

Pr´ eville-Ratelle–Viennot: Introduced the ν-Tamari lattice on lattice paths weakly above ν. Covering relation: They also have nice enumerative and geometric properties.

Fang–Pr´ eville-Ratelle. The enumeration of generalized Tamari intervals. European Journal of Combinatorics 61, 2017. C.–Padrol–Sarmiento. Geometry of ν-Tamari lattices in types A and B. arXiv:1611.09794, 2016.

First theorem

Theorem 1

The Hasse diagram of the ν-Tamari lattice is the facet adjacency graph of a well chosen subword complex .

This generalizes a known result by Woo (2004), Pilaud–Pocchiola (2010), Stump (2010), and Stump-Serrano (2010) in the classical case.

Subword complexes

W = Sn+1 group of permutations of [n + 1] S = {s1, . . . , sn} the set of simple generators si = (i i + 1) Q = (q1, . . . , qm) a word in S π ∈ W

Subword complexes

W = Sn+1 group of permutations of [n + 1] S = {s1, . . . , sn} the set of simple generators si = (i i + 1) Q = (q1, . . . , qm) a word in S π ∈ W

Definition (Knutson–Miller, 2004)

The subword complex ∆(Q, π) is the simplicial complex whose faces ← → subwords P of Q such that Q \ P contains a reduced expression of π

Knutson–Miller. Gr¨

• bner geometry of Schubert polynomials. Ann. Math., 161(3), ’05

Knutson–Miller. Subword complexes in Coxeter groups. Adv. Math., 184(1), ’04

Subword complexes - Example modify s3

In type A2: W = S3, S = {s1, s2} = {(1 2), (2 3)}

Subword complexes - Example modify s3

In type A2: W = S3, S = {s1, s2} = {(1 2), (2 3)} Q = ( s1,s2,s1,s2,s1 ) q1,q2,q3,q4,q5 and π = [3 2 1]

Subword complexes - Example modify s3

In type A2: W = S3, S = {s1, s2} = {(1 2), (2 3)} Q = ( s1,s2,s1,s2,s1 ) q1,q2,q3,q4,q5 and π = [3 2 1] ∆(Q, π) is isomorphic to q1 q2 q3 q4 q5

Subword complexes - Example modify s3

In type A2: W = S3, S = {s1, s2} = {(1 2), (2 3)} Q = ( , ,s1,s2,s1 ) q1,q2, , , and π = [3 2 1] = s1s2s1 ∆(Q, π) is isomorphic to q1 q2 q3 q4 q5

Subword complexes - Example modify s3

In type A2: W = S3, S = {s1, s2} = {(1 2), (2 3)} Q = ( s1, , ,s2,s1 ) ,q2,q3, , and π = [3 2 1] = s1s2s1 ∆(Q, π) is isomorphic to q1 q2 q3 q4 q5

Subword complexes - Example modify s3

In type A2: W = S3, S = {s1, s2} = {(1 2), (2 3)} Q = ( s1,s2, , ,s1 ) , ,q3,q4, and π = [3 2 1] = s1s2s1 ∆(Q, π) is isomorphic to q1 q2 q3 q4 q5

Subword complexes - Example modify s3

In type A2: W = S3, S = {s1, s2} = {(1 2), (2 3)} Q = ( s1,s2,s1, , ) , , ,q4,q5 and π = [3 2 1] = s1s2s1 ∆(Q, π) is isomorphic to q1 q2 q3 q4 q5

Subword complexes - Example modify s3

In type A2: W = S3, S = {s1, s2} = {(1 2), (2 3)} Q = ( ,s2,s1,s2, ) q1, , , ,q5 and π = [3 2 1] = s1s2s1 = s2s1s2 ∆(Q, π) is isomorphic to q1 q2 q3 q4 q5

Subword complexes - Example modify s3

In type A2: W = S3, S = {s1, s2} = {(1 2), (2 3)} Q = ( s1,s2,s1,s2,s1 ) q1,q2,q3,q4,q5 and π = [3 2 1] = s1s2s1 = s2s1s2 ∆(Q, π) is isomorphic to q1 q2 q3 q4 q5

The subword complex result

Theorem 1

The Hasse diagram of the ν-Tamari lattice is the facet adjacency graph of a well chosen subword complex ∆(Qν, πν).

s1 s2 s2 s5 s3 s4 s3 s3 s4 s4

1 4 3 5 2 6

Qν = (s3, s2, s1, s4, s3, s2, s4, s3, s5, s4) πν = [1, 4, 3, 5, 2, 6]

The subword complex result

These objects keep showing up in independent places:

Serrano–Stump. Maximal fillings of moon polyominoes, simplicial complexes, and Schubert polynomials. Electron. J. Combin., 19(1), 2012. M´ esz´

• aros. Root polytopes, triangulations, and the subdivision algebra. I. Trans.
• Amer. Math. Soc., 363(8), 2011.

Escobar–M´ esz´

• aros. Subword complexes via triangulations of root polytopes.

arXiv:1502.03997.

s1 s2 s2 s5 s3 s4 s3 s3 s4 s4

1 4 3 5 2 6

They are special but still some what mysterious.

Facets and ν-trees

The facets of ∆(Qν, πν) are given by ν-trees. Two facets are adjacent ↔ the trees are related by rotation. s2s3s2s4 = [1, 4, 3, 5, 2, 6] s2 s3 s2 s4 s3s2s3s4 = [1, 4, 3, 5, 2, 6] s2 s3 s3 s4

rotation

Facets and ν-trees

The facets of ∆(Qν, πν) are given by ν-trees. Two facets are adjacent ↔ the trees are related by rotation. s2s3s2s4 = [1, 4, 3, 5, 2, 6] s2 s3 s2 s4 s3s2s3s4 = [1, 4, 3, 5, 2, 6] s2 s3 s3 s4

rotation

ν-tree: (Serrano–Stump) Maximal sets of lattice points above ν avoiding north-east increasing chains p, q such that pq is above ν. (This talk) some “maximal” binary trees fitting above ν.

The rotation lattice of ν-trees

Theorem 1 follows from:

Theorem 2

The ν-Tamari lattice is isomorphic to the rotation lattice on ν-trees.

The rotation lattice of ν-trees

Theorem 1 follows from:

Theorem 2

The ν-Tamari lattice is isomorphic to the rotation lattice on ν-trees.

2 4 1 5 6 3 2 4 1 5 6 3 Right flushing Left flushing

The lattice of ν-bracket vectors

The meet and join: very simple on ν-trees.

The lattice of ν-bracket vectors

The meet and join: very simple on ν-trees.

Theorem 3

The ν-Tamari lattice is isomorphic to the lattice of ν-bracket vectors under componentwise order.

1 1 1 1 1 1 1 2 2 1 2 1 1 1 1 1 1 1 1 2 2 2 2 2 2 2 2 2 2 2 2 2

(0,0,1,1,1,2) (1,0,1,1,1,2) (2,0,1,1,1,2) (2,0,2,1,1,2) (2,0,2,2,1,2) (0,0,2,2,1,2) (0,0,2,1,1,2)

b(T) = read y-coordinates of the nodes in in-order.

The lattice of ν-bracket vectors

ν-bracket vectors are easily characterized. Their meet is obtained by taking componentwise minimum.

Corollary

Simple proof of the lattice property.

1 1 1 1 1 1 1 2 2 1 2 1 1 1 1 1 1 1 1 2 2 2 2 2 2 2 2 2 2 2 2 2

(0,0,1,1,1,2) (1,0,1,1,1,2) (2,0,1,1,1,2) (2,0,2,1,1,2) (2,0,2,2,1,2) (0,0,2,2,1,2) (0,0,2,1,1,2)

In summary

1 1 1 1 1 1 1 2 2 1 2 1 1 1 1 1 1 1 1 2 2 2 2 2 2 2 2 2 2 2 2 2

(0,0,1,1,1,2) (1,0,1,1,1,2) (2,0,1,1,1,2) (2,0,2,1,1,2) (2,0,2,2,1,2) (0,0,2,2,1,2) (0,0,2,1,1,2)

Thm 1 & Thm 2 Thm 3 & Cor

Multi ν-Tamari complexes

For k ≥ 1, define the (k, ν)-Tamari complex faces ↔ sets of points above ν avoiding (k + 1)-north-east incr. chains.

◮ ν = (NE)n: simplicial multiassociahedron ∆n+2,k.

Conjectured to be realizable as a polytope (Jonsson 2004).

◮ k = 1, ν without consecutive north steps: facet adjacency graph =

edge graph of a polytopal subdivision of an associahedron.

Question

Is the facet adjacency graph of the (k, ν)-Tamari complex the edge graph

• f a polytopal subdivision of a multiassociahedron?

Multi ν-Tamari complexes

Proposition

Let m ≥ k and ν = (NE m)k+1. The facet adjacency graph Gk,ν of the Fuss-Catalan (k, ν)-Tamari complex is the edge graph of a polytopal subdivision of the multi-associahedron ∆2k+2,k.

k = 2 and ν = (NE 5)3 k = 3 and ν = (NE 5)4

∆2k+2,k: a k-dimensional simplex Subdivision: staircase subdivision of its (m − k + 1) dilation.

My birthday present! Is this true in general?

Merci!